An algorithm for finding all the<i>k</i>-components of a digraph

Makino, S.

doi:10.1080/00207168808803645

Cited by 13 publications

(10 citation statements)

References 10 publications

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“…There is a different line of work on "k-connected components" that, as far as I can see, is completely unrelated to ours. There, k-connected components are simply defined as maximal k-connected subgraphs (see, for example, [12,15,14]). This leads to completely different decompositions.…”

Section: Related Workmentioning

confidence: 99%

Quasi-4-connected components

Grohe

2017

Descriptive Complexity, Canonisation, and Definable Graph Structure Theory

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We introduce a new decomposition of a graphs into quasi-4-connected components, where we call a graph quasi-4-connected if it is 3-connected and it only has separations of order 3 that remove a single vertex. Moreover, we give a cubic time algorithm computing the decomposition of a given graph.Our decomposition into quasi-4-connected components refines the well-known decompositions of graphs into biconnected and triconnected components. We relate our decomposition to Robertson and Seymour's theory of tangles by establishing a correspondence between the quasi-4-connected components of a graph and its tangles of order 4.

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Section: Related Workmentioning

confidence: 99%

Quasi-4-connected components

Grohe

2017

Descriptive Complexity, Canonisation, and Definable Graph Structure Theory

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“…For undirected graphs, it has been known for over 40 years how to compute the analogous notions (bridges, articulation points, 2-edge-and 2-vertex-connected components) in linear time, by simply using depth first search [38]. In the case of digraphs, however, the same problems revealed to be much more challenging: although these problems have been investigated for quite a long time (see, e.g., [13,34,36]), obtaining fast algorithms for 2-edge and 2-vertex connectivity for digraphs has been an elusive goal for many years. Indeed, it has been shown only recently that all strong bridges and strong articulation points of a digraph can be computed in linear time [27].…”

Section: Introductionmentioning

confidence: 99%

Strong Connectivity in Directed Graphs under Failures, with Applications

Georgiadis¹,

Italiano²,

Parotsidis³

2020

SIAM J. Comput.

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In this paper, we investigate some basic connectivity problems in directed graphs (digraphs). Let G be a digraph with m edges and n vertices, and let G \ e (resp., G \ v) be the digraph obtained after deleting edge e (resp., vertex v) from G. As a first result, we show how to compute in O(m + n) worst-case time: • The total number of strongly connected components in G \ e (resp., G \ v), for all edges e (resp., for all vertices v) in G. • The size of the largest and of the smallest strongly connected components in G \ e (resp., G \ v), for all edges e (resp., for all vertices v) in G. Let G be strongly connected. We say that edge e (resp., vertex v) separates two vertices x and y, if x and y are no longer strongly connected in G \ e (resp., G \ v). As a second set of results, we show how to build in O(m + n) time O(n)-space data structures that can answer in optimal time the following basic connectivity queries on digraphs: • Report in O(n) worst-case time all the strongly connected components of G \ e (resp., G \ v), for a query edge e (resp., vertex v). • Test whether an edge or a vertex separates two query vertices in O(1) worst-case time. • Report all edges (resp., vertices) that separate two query vertices in optimal worst-case time, i.e., in time O(k), where k is the number of separating edges (resp., separating vertices). (For k = 0, the time is O(1)). All our bounds are tight and are obtained with a common algorithmic framework, based on a novel compact representation of the decompositions induced by the 1-connectivity (i.e., 1-edge and 1-vertex) cuts in digraphs, which might be of independent interest. With the help of our data structures we can design efficient algorithms for several other connectivity problems on digraphs and we can also obtain in linear time a strongly connected spanning subgraph of G with O(n) edges that maintains the 1-connectivity cuts of G and the decompositions induced by those cuts.

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“…For undirected graphs it has been known for over 40 years how to compute the 2-edge-and 2-vertexconnected components in linear time [36]. In the case of digraphs, however, only O(mn) algorithms were known (see e.g., [27,28,31,33]). It was shown only recently how to compute the 2-edge-and 2-vertex-connected blocks in linear time [14,15], and the best current bound for computing the 2-edge-and the 2-vertex-connected components is O(n 2 ) [20].…”

Section: Introductionmentioning

confidence: 99%

Incremental Strong Connectivity and 2-Connectivity in Directed Graphs

Georgiadis

Italiano

Parotsidis

2018

LATIN 2018: Theoretical Informatics

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In this paper, we present new incremental algorithms for maintaining data structures that represent all connectivity cuts of size one in directed graphs (digraphs), and the strongly connected components that result by the removal of each of those cuts. We give a conditional lower bound that provides evidence that our algorithms may be tight up to a sub-polynomial factors. As an additional result, with our approach we can also maintain dynamically the 2-vertex-connected components of a digraph during any sequence of edge insertions in a total of O(mn) time. This matches the bounds for the incremental maintenance of the 2-edge-connected components of a digraph.

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An algorithm for finding all thek-components of a digraph

Cited by 13 publications

References 10 publications

Quasi-4-connected components

Quasi-4-connected components

Strong Connectivity in Directed Graphs under Failures, with Applications

Incremental Strong Connectivity and 2-Connectivity in Directed Graphs

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